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Spinning Mellin Bootstrap
Massimo Taronna
Princeton University
Mostly based on: 1702.08619, 1708.08404, 1804.09334 & to appear
w. Charlotte Sleight
Feynman diagrams quickly
become unmanageable,
however final result of the
resummation of many
complicated diagram is very
often simple
Use symmetry & Quantum Mechanics
to find the answer directly
Challenges in Analytic Bootstrap
The basic idea is to bypass Feynman diagram (bulk or boundary) computation
and just impose:
• use Mellin space to uncover explicitly inversion formulas!
• Clarify the role of AdS/CFT at tree level (kinematic transform)
• Demystify “holographic reconstruction”: equivalence between Noether
procedure and bootstrap at tree-level
• Revisit the role of differential operators to generate spinning blocks in terms
of higher-spin generators
• …
Goals (if time permits):
Inversion formulas in S-matrix
Inversion formula are standard tools in S-matrix theory
Partial waves
(fixed by isometries – kinematics)
Obtain the spin J coefficient directly from S-matrix
Study the above problem as a function of spin: “continuous spin”
Inversion formulas & Bulk locality
What is the interpretation of analyticity in spin?
Inversion = Cauchy
Euclidean Integral
around origin (l integer)
brunch-cut
analytic for continuous l:
continuous spin
Coefficients form an infinite family and have to be
varied analytically
Inversion formulas in CFT
Inversion formula are standard tools in Harmonic analysis: diagonalize Casimir
in a way that preserves self-adjointness
Self-adjointness requires that f and g are single valued functions (in Euclidean kinematics)
An orthogonal basis of eigenfunctions of the Casimir can be found in terms of
conformal partial waves
Conformal block Shadow
Conformal Partial Wave (CPW)
Inversion formulas
Inversion formula are standard tools in Harmonic analysis: diagonalize Casimir
in a way that preserves self-adjointness
Ortogonality & completeness requires that Delta is on the principal series:
The above Euclidean formula is the basis for recent developments of analytic bootstrap
[Alday et al., Caron-Huot, …]
Regge Limit:
Inversion formulas
Toy example of analytic continuation in S-matrix theory (d=2)
Gegenbauer polynomials in d=2
Close the contour to the center if J big enough to
obtain an integral over a discontinuity
Similar steps in CFT allow to express the function c in terms of the double-
discontinuity of the correlator
The CPW gets analytically continued into a conformal block with spin and dimension
interchanged (analyticity in spin/continuous spin)
[Caron-Huot, 2017]
Inversion formulas & Bulk locality
Inversion formula tells us that the only free parameters are the first 2
All other terms have to resum to reproduce the discontinuity of the amplitude (EFT).
[Sleight & M.T. 2017]
Contact terms beyond the first few must resum to 1/Box (EFT)
Mellin space
So far all integral formulas we wrote required careful analysis of the conformal
integrals involved (gauge fixing etc…)
is there a way to make manifest these orthogonality properties?
Standard 4pt conformal integral
Symanzik star formula allows to evaluate these integral in terms of a Mellin representation
Mack Polynomials
Mellin space
Mack polynomials encode conformal partial waves in terms of degree l polynomials in
analogues of Mandelstam variables
Projecting out the shadow poles is straightforward:
For each primary operator we have an infinite series of poles:
Physical pole
Descendants pole
[Fitzpatrick & Kaplan 2011]
Mellin space
Orthogonality of CPWs becomes manifest in Mellin space:
The kinematic polynomials turn out to be Continuous Hahn polynomials (3F2)
Position space orthogonality becomes manifest in Mellin space!
What about spinning external legs?
[Costa et al.]
Spinning Correlators
Spinning correlators require to introduce tensorial structures
3pt functions can be decomposed in terms of monomials:
Conformal symmetry allows to reconstruct the correlator from a subset of the structures
Spinning CPWs
The definition of CPWs given in the scalar case is very general
The above integral can be explicitly performed in Mellin space but without a
guiding principle its form does not show any structure
Orthogonality is not manifest because it involves a delicate interplay
between different tensor structures…
We will argue that a key guiding principle lies in the bulk-to-boundary map:
(AdS/CFT)
Tree level AdS/CFT ~ momentum space
AdSd+1 CFTd
Single-trace
Primary operator
CFTd
(E)AdSd+1
Fields
Single-particle state
spinscaling dim.
What is the bulk dual (position space version) of a CPW?[Ferrara, Grillo, Gatto, Todorov, Fronsdal...]
“Momentum space” for AdS
Harmonic functions factorise into bulk-to-boundary propagators:
[Massive fields: Costa et al.`14, Massless: Bekaert et al. `14; Sleight, M.T. `17]
[Leonhardt, Manvelyan, Rühl `03; Costa et al.`14]
Expand in basis of bi-tensorial harmonic functions (analogue of plane waves):
divergence-less trace-less
Bulk-to-bulk propagators:
At tree-level, diagrams factorise into lower-point trees,
which are connected via conformal integration over the boundary:
Bulk-to-bulk propagators:
No AdS/CFT assumption but only kinematical rewriting!
Expand in basis of bi-tensorial harmonic functions (analogue of plane waves):
divergence-less trace-less
“Momentum space” for AdS
[Mück et al.; Freedman et al. `98]
Standard Trick: Reduce integral over AdS to its scalar seed
“Fourier transforming” 3pt vertices
Result takes the form:
Spinning tree level 3pt diagrams
The above problem suggest a new basis for 3pt CFT structures:
Linear Map for any cubic
coupling
We can holographically reconstruct each basis element
Weight Shifting OperatorsCubic couplings induce deformations of gauge transformations and gauge symmetries
Explicit classification known in AdS/CFT
The commutator of two gauge transformations closes to the lowest order automatically: extract gauge bracket (field independent)
The deformation of gauge transformations are the most general conformal
differential operators that can be written down!
(a) Labels the
possible
differential
operators
[Joung & MT ’13]
Weight Shifting OperatorsClosure, Jacobi, covariance of cubic couplings can be explicitly written down in terms of 6j symbols of the conformal group:
One obtains crossing relations for HS transformations
6j symbol
Weight Shifting OperatorsClosure, Jacobi, covariance of cubic couplings can be explicitly written down in terms of 6j symbols of the conformal group:
Noether procedure for cubic vertices at quartic order:
Many solutions are
known: type An, Bn,…
Infinite number of equations but finitely many
terms for each equation fix local vertices uniquely
Going to Mellin Space
The coupling itself knows everything of the differential operator
Everything is reduced to a single scalar integral!
Orthogonality of conformal blocks can be read off from the leading pole,
e.g.:
Remarkable Fact: factorization of l dependence from external spin dependence!!
Inversion formulas manifest in terms of the Continuous Hahn polynomial
Applications
• Crossing Kernels
• Large N fixed points
• Wilson-Fisher
Crossing Kernels
Crossing Kernel/Racah Coefficient/6j symbol
Arbitrary exchanged spin (single structure):
Tetrahedral sum
[C. Sleight & M.T.]
The first step is to extract the leading order OPE
A simple test for inversion formula but we need to go to Mellin space…
This integral is
divergent…
Mean Field Theory
Mean Field TheoryThe first step is to extract the leading order OPE
A simple test for inversion formula but we need to go to Mellin space…
This integral is
divergent… (as for HS
theory in flat space)Can still define it as a distribution upon considering it as a
functional on an appropriate space of functions
The Mellin transform of Wick-contractions is a delta-function distribution
[(a lot of papers in feynman diagram community), M.T. 2016; Bekaert et al. 2016]
O(N) modelThe O(N) model is not much different than MFT
The above conformal block expansion can be arranged in twist block expansions
N.B. The above sum are not uniformly convergent:
Sum over spin must reproduce singularities in the crossed channels…
Anomalous DimensionsThe simplest external scalar case:
Subleading twist operators
Leading twist operators
We can easily obtain the
result for scalar double-trace
deformations
Anomalous DimensionsThe simplest external scalar case:
Subleading twist operators
Leading twist operators
We obtain explicit expressions for all subleading twist double trace operators
[OO]1,0
On the real axis the dimension of the CPW in t, u channel. The bar is the dimension of the external legs
[OO]2,0
On the real axis the dimension of the CPW in t, u channel. The bar is the dimension of the external legs
Wilson-FisherThe simplest external scalar case:
Subleading twist operators
Leading twist operators
We obtain a closed formula for arbitrary l and n:
The above result applies to the WF-fixed point with:
Large spin behavior same independently on n:
J0J0MFT:
First the OPE:
J0J0MFT:
And then
anomalous
dimensions:
Outlook• We barely scratched the surface of a remarkable hidden structure behind
CFT conformal blocks with spinning external and internal legs!
• Mellin space makes manifest inversion formulas and reduces them to
finite dimensional linear algebra
• Lesson: The bulk to boundary explicit map of arXiv:1702.08619 can
teach us a lot about the spinning bootstrap and it is the analogues of
momentum space for flat space HS correlators
• Analyticity in spin sets the convergence rate of quartic interactions (EFT
& 1/Box)
Thank you!